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March  2013, 33(3): 1061-1076. doi: 10.3934/dcds.2013.33.1061

Boundary values of the resolvent of Schrödinger hamiltonians with potentials of order zero

J. Cruz-Sampedro 1,

1. 

Departamento de Ciencias Básicas, UAM-A, Av. San Pablo 180, Col. Reynosa, México D.F. 02200, Mexico

Received  April 2011 Revised  October 2011 Published  October 2012

Let $H=-\Delta +V$ be a Schrödinger hamiltonian acting on $L^2(\mathbb{R}^n)$, $n\geq 2$, where $V$ a potential of order zero plus a short-range perturbation. In this work we investigate the behavior of the resolvent $R(z)=(H-z)^{-1}$ of $H$ as Im$\,z \downarrow 0$, at high energies and in the framework of Besov spaces $B(\mathbb{R}^n)$. For $\lambda_0>0$ sufficiently large and $\lambda\geq\lambda_0$, we show that there exists a linear operator $R(\lambda+i0)$ such that $R(\lambda+i\epsilon)$ converges to $R(\lambda+i0)$ as $\epsilon\downarrow 0$, strongly in $\mathcal{L}(L^{2, s}(\mathbb{R}^n),L^{2,-s}(\mathbb{R}^n))$, $s>1/2$, and weakly in $\mathcal{L}(B(\mathbb{R}^n),B^*(\mathbb{R}^n))$. We achieve this through a Mourre-estimate strategy.
Keywords: order-zero potentials, Schrödinger hamiltonian, Besov spaces..
Mathematics Subject Classification: Primary: 35P25, 81U99, 47A40; Secondary: 46C9.
Citation: J. Cruz-Sampedro. Boundary values of the resolvent of Schrödinger hamiltonians with potentials of order zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1061-1076. doi: 10.3934/dcds.2013.33.1061
References:
[1]

S. Agmon, Spectral properties of Schrödinger operators and scattering theory,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 151. Google Scholar

[2]

S. Agmon, J. Cruz and I. Herbst, Generalized Fourier transform for Schrödinger operators with potentials of order zero,, Journal of Functional Analysis, 167 (1999), 345. doi: 10.1006/jfan.1999.3432. Google Scholar

[3]

S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics,, Journal D'Analyse Mathématique, 30 (1976). Google Scholar

[4]

G. Barles, On eikonal equations associated with Schrödinger operators with nonspherical radiation conditions,, Commun. in Partial Differential Equations, 12 (1987), 263. Google Scholar

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J. Cruz-Sampedro, The eikonal equation and a class of Schrödinger-like operators,, Submitted 2011., (2011). Google Scholar

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J. Dereziński and C. Gérard, "Scattering Theory of Classical and Quantum $N$-Particle Systems,", Springer, (1997). Google Scholar

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W. Jäger, Über das Dirichletsche Auβenraumproblem fü die Schwingungsgleichung,, Math. Zeitschr, 95 (1967), 299. doi: 10.1007/BF01111082. Google Scholar

[8]

A. Jensen and P. Perry, Commutator methods and Besov space estimates for Schrödinger operators,, J. Operator Theory, 14 (1985), 181. Google Scholar

[9]

D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators,, Ann. Math., 121 (1985), 463. doi: 10.2307/1971205. Google Scholar

[10]

E. Mourre, Absence of singular continuous spectrum for certain self-adjoint operators,, Commun. Math. Phys., 78 (): 391. doi: 10.1007/BF01942331. Google Scholar

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P. Perry, I. Sigal and B. Simon, Spectral analysis of $N$-body Schrödinger operators,, Ann. Math., 114 (1981), 519. doi: 10.2307/1971301. Google Scholar

[12]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics, III Sacattering Theory,", New York, (1979). Google Scholar

[13]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators,", Academic Press, (1978). Google Scholar

[14]

Y. Saitō, Schrödinger operators with a nonspherical radiation condition,, Pacific Journal of Mathematics, 126 (1987), 331. Google Scholar

[15]

I. Sigal, "Scattering Theory for Many-Body Quantum Mechanical Systems,", Lecture Notes in Mathematics 1011, (1011). Google Scholar

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G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus,, (French) Ann. Inst. Fourier (Grenoble), 15 (1965), 189. Google Scholar

show all references

References:
[1]

S. Agmon, Spectral properties of Schrödinger operators and scattering theory,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 151. Google Scholar

[2]

S. Agmon, J. Cruz and I. Herbst, Generalized Fourier transform for Schrödinger operators with potentials of order zero,, Journal of Functional Analysis, 167 (1999), 345. doi: 10.1006/jfan.1999.3432. Google Scholar

[3]

S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics,, Journal D'Analyse Mathématique, 30 (1976). Google Scholar

[4]

G. Barles, On eikonal equations associated with Schrödinger operators with nonspherical radiation conditions,, Commun. in Partial Differential Equations, 12 (1987), 263. Google Scholar

[5]

J. Cruz-Sampedro, The eikonal equation and a class of Schrödinger-like operators,, Submitted 2011., (2011). Google Scholar

[6]

J. Dereziński and C. Gérard, "Scattering Theory of Classical and Quantum $N$-Particle Systems,", Springer, (1997). Google Scholar

[7]

W. Jäger, Über das Dirichletsche Auβenraumproblem fü die Schwingungsgleichung,, Math. Zeitschr, 95 (1967), 299. doi: 10.1007/BF01111082. Google Scholar

[8]

A. Jensen and P. Perry, Commutator methods and Besov space estimates for Schrödinger operators,, J. Operator Theory, 14 (1985), 181. Google Scholar

[9]

D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators,, Ann. Math., 121 (1985), 463. doi: 10.2307/1971205. Google Scholar

[10]

E. Mourre, Absence of singular continuous spectrum for certain self-adjoint operators,, Commun. Math. Phys., 78 (): 391. doi: 10.1007/BF01942331. Google Scholar

[11]

P. Perry, I. Sigal and B. Simon, Spectral analysis of $N$-body Schrödinger operators,, Ann. Math., 114 (1981), 519. doi: 10.2307/1971301. Google Scholar

[12]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics, III Sacattering Theory,", New York, (1979). Google Scholar

[13]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators,", Academic Press, (1978). Google Scholar

[14]

Y. Saitō, Schrödinger operators with a nonspherical radiation condition,, Pacific Journal of Mathematics, 126 (1987), 331. Google Scholar

[15]

I. Sigal, "Scattering Theory for Many-Body Quantum Mechanical Systems,", Lecture Notes in Mathematics 1011, (1011). Google Scholar

[16]

G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus,, (French) Ann. Inst. Fourier (Grenoble), 15 (1965), 189. Google Scholar

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